A recent question on puzzling relied on hyper-sphere having maximal area at dimension 7.
Which is shown here.
What I can't shake off is the feeling that one can't compare the surface of unit spheres of different dimensions. It seems the units would be totally unrelated. A sphere has a surface in squared units, a circle has a length in units and a 4-d hyper-sphere has an area in cubic units (typically a volume, for us in 3d).
But the fact that someone went to the effort of making this wikipedia graph implies there's some usefulness or value there. Or even just that the values are comparable in some way.
What am I missing?
Best Answer
You cannot compare measures itself, but you can compare coefficients before $r^n$ in a corresponding formula.
In other words you can compare how large the volume of unit n-sphere is in comparison to the volume of unit cube. I will leave the usefulness of this aside.